Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application.

In drug development, molecular docking methods aim at characterizing the binding of a drug-like molecule to a protein. In a typical drug development process, a docking task is repeated millions of time, which makes optimization efforts essential. In particular, modern heterogeneous architectures, such as GPUs, allow for significant acceleration opportunities. AutoDock-GPU, a state-of-the-art GPU-accelerated molecular docking software, estimates the geometrical conformation of a docked ligand-protein complex by minimizing an energy-based scoring function. Our profiling results indicated that a reduction operation, which is performed several millions times in a single docking run, limits performance in AutoDock-GPU. Thus, we proposed a method to accelerate the block-level sum reduction of four-element vectors by using matrix operations. We implemented our method to make use of the high throughput capabilities offered by NVIDIA Tensor Cores to perform matrix operations. We evaluated our approach by designing a simple benchmark, and achieved a 4 to 7-fold runtime improvement compared to the original method. We then integrated our reduction operation into AutoDock-GPU and evaluated it on multiple chemical complexes on three GPUs. This evaluation allowed to assess the possibility to use half-precision reduction operations in parts of AutoDock-GPU code, without detrimental effects on the simulation result. In addition, our implementation achieved an average 27% improvement on the overall docking time during a real-world docking run. / Vid läkemedelsutveckling syftar molekylär dockningsmetoder till att karakterisera bindningen av en läkemedelsliknande molekyl till ett protein. I en typisk läkemedelsutvecklingsprocess upprepas en dockinguppgift miljontals gånger, vilket gör optimeringsinsatser nödvändiga. Framför allt moderna heterogena arkitekturer som GPU:er ger betydande accelerationsmöjligheter. AutoDock-GPU, en modern GPU-accelererad programvara för molekylär dockning, uppskattar den geometriska konformationen hos ett ligand-protein-komplex genom att minimera en energibaserad poängsättningsfunktion. Våra profileringsresultat visade att en reduktionsoperation, som utförs flera miljoner gånger i en enda dockningskörning, begränsar prestandan i AutoDock-GPU. Vi har därför föreslagit en metod för att accelerera summareduktionen på blocknivå av vektorer med fyra element med hjälp av matrisoperationer. Vi implementerade vår metod för att utnyttja den höga genomströmningskapacitet som erbjuds av NVIDIA Tensor Cores för att utföra matrisoperationer. Vi utvärderade vårt tillvägagångssätt genom att utforma ett enkelt testfall och uppnådde en 4- till 7-faldig förbättring av körtiden jämfört med den ursprungliga metoden. Vi integrerade sedan vår reduktionsoperation i AutoDock-GPU och utvärderade den på flera kemiska komplex på tre GPU:er. Denna utvärdering lät oss bedöma möjligheten att använda reduktionsoperationer med halvprecision i delar av AutoDock-GPU-koden, utan negativa effekter på simuleringsresultatet. Dessutom uppnådde vår version en genomsnittlig förbättring på 27% av den totala dockningstiden under en riktig dockningskörning.

Les méthodes de réseau de tenseurs (TN) ont été un ingrédient essentiel des progrès de la physique de la matière condensée et ont récemment suscité l'intérêt de la communauté de l'apprentissage automatique pour leur capacité à représenter de manière compacte des objets de très grande dimension. Les méthodes TN peuvent par exemple être utilisées pour apprendre efficacement des modèles linéaires dans des espaces de caractéristiques exponentiellement grands [1]. Dans ce manuscrit, nous dérivons des limites supérieures et inférieures sur la VC-dimension et la pseudo-dimension d'une grande classe de Modèles TN pour la classification, la régression et la complétion . Nos bornes supérieures sont valables pour les modèles linéaires paramétrés par structures TN arbitraires, et nous dérivons des limites inférieures pour les modèles de décomposition tensorielle courants (CP, Tensor Train, Tensor Ring et Tucker) montrant l'étroitesse de notre borne supérieure générale. Ces résultats sont utilisés pour dériver une borne de généralisation qui peut être appliquée à la classification avec des matrices de faible rang ainsi qu'à des classificateurs linéaires basés sur l'un des modèles de décomposition tensorielle couramment utilisés. En corollaire de nos résultats, nous obtenons une borne sur la VC-dimension du classificateur basé sur le matrix product state introduit dans [1] en fonction de la dimension de liaison (i.e. rang de train tensoriel), qui répond à un problème ouvert répertorié par Cirac, Garre-Rubio et Pérez-García [2]. / Tensor network (TN) methods have been a key ingredient of advances in condensed matter physics and have recently sparked interest in the machine learning community for their ability to compactly represent very high-dimensional objects. TN methods can for example be used to efficiently learn linear models in exponentially large feature spaces [1]. In this manuscript, we derive upper and lower bounds on the VC-dimension and pseudo-dimension of a large class of TN models for classification, regression and completion. Our upper bounds hold for linear models parameterized by arbitrary TN structures, and we derive lower bounds for common tensor decomposition models (CP, Tensor Train, Tensor Ring and Tucker) showing the tightness of our general upper bound. These results are used to derive a generalization bound which can be applied to classification with low-rank matrices as well as linear classifiers based on any of the commonly used tensor decomposition models. As a corollary of our results, we obtain a bound on the VC-dimension of the matrix product state classifier introduced in [1] as a function of the so-called bond dimension (i.e. tensor train rank), which answers an open problem listed by Cirac, Garre-Rubio and Pérez-García [2].

The General-Purpose GPU (GPGPU) has emerged as the predominant computing device for extensive parallel workloads in the fields of Artificial Intelligence (AI) and Scientific Computing, primarily owing to its adoption of the Single Instruction Multiple Thread architecture, which not only provides a wealth of thread context but also effectively hide the latencies exposed in the single threads executions. As computational demands have evolved, modern GPGPUs have incorporated specialized matrix engines, e.g., NVIDIA’s Tensor Core (TC), in order to deliver substantially higher throughput for dense matrix computations compared with traditional scalar or vector architectures. Beyond mere throughput, energy efficiency is a pivotal concern in GPGPU computing. The register file is the largest memory structure on the GPGPU die and typically accounts for over 20% of the dynamic power consumption. To enhance energy efficiency, GPGPUs incorporate a technique named register caching borrowed from the realm of CPUs. Register caching captures temporal locality among register operands to reduce energy consumption within a 2- level register file structure. The presence of TC raises new challenges for Register Cache (RC) design, as each matrix instruction applies intensive operand delivering traffic on the register file banks. In this study, we delve into the RC design trade-offs in GPGPUs. We undertake a comprehensive exploration of the design space, encompassing a range of workloads. Our experiments not only reveal the basic design considerations of RC but also clarify that conventional caching strategies underperform, particularly when dealing with TC computations, primarily due to poor temporal locality and the substantial register operand traffic involved. Based on these findings, we propose an enhanced caching strategy featuring a look-ahead allocation policy to minimize unnecessary cache allocations for the destination register operands. Furthermore, to leverage the energy efficiency of Tensor Core computing, we highlight an alternative instruction scheduling framework for Tensor Core instructions that collaborates with a specialized caching policy, resulting in a remarkable reduction of up to 50% in dynamic energy consumption within the register file during Tensor Core GEMM computations. / Den allmänna ändamålsgrafikprocessorn (GPGPU) har framträtt som den dominerande beräkningsenheten för omfattande parallella arbetsbelastningar inom områdena för artificiell intelligens (AI) och vetenskaplig beräkning, huvudsakligen tack vare dess antagande av arkitekturen för enkel instruktion, flera trådar (Single Instruction Multiple Thread), vilket inte bara ger en mängd trådcontext utan också effektivt döljer de latenser som exponeras vid enskilda trådars utförande. När beräkningskraven har utvecklats har moderna GPGPU:er inkorporerat specialiserade matrismotorer, t.ex., NVIDIAs Tensor Core (TC), för att leverera avsevärt högre genomströmning för täta matrisberäkningar jämfört med traditionella skalär- eller vektorarkitekturer. Bortom endast genomströmning är energieffektivitet en central oro inom GPGPUberäkning. Registerfilen är den största minnesstrukturen på GPGPU-dien och svarar vanligtvis för över 20% av den dynamiska effektförbrukningen För att förbättra energieffektiviteten inkorporerar GPGPU:er en teknik vid namn registercachning, lånad från CPU-världen. Registercachning fångar temporal lokalitet bland registeroperanderna för att minska energiförbrukningen inom en 2-nivåers registerfilstruktur. Närvaron av TC innebär nya utmaningar för Register Cache (RC)-design, eftersom varje matrisinstruktion genererar intensiv operandleverans på registerfilbankarna. I denna studie fördjupar vi oss i RC-designavvägandena i GPGPU:er. Vi genomför en omfattande utforskning av designutrymmet, som omfattar olika arbetsbelastningar. Våra experiment avslöjar inte bara de grundläggande designövervägandena för RC utan klargör också att konventionella cachestrategier underpresterar, särskilt vid hantering av TC-beräkningar, främst på grund av dålig temporal lokalitet och den betydande trafiken med registeroperand. Baserat på dessa resultat föreslår vi en förbättrad cachestrategi med en look-ahead-alloceringspolicy för att minimera onödiga cacheallokeringar för destinationens registeroperand. Dessutom, för att dra nytta av energieffektiviteten hos Tensor Core-beräkning, belyser vi en alternativ instruktionsplaneringsram för Tensor Core-instruktioner som samarbetar med en specialiserad cachelayout, vilket resulterar i en anmärkningsvärd minskning av upp till 50% i dynamisk energiförbrukning inom registerfilen under Tensor Core GEMM-beräkningar.

In dieser Arbeit wird eine unrestricted Coupled-Cluster CC2 Response-Methode für die Berechnung von Eigenschaften erster und zweiter Ordnung, mit dem elektronischen g-Tensor als Schwerpunkt, präsentiert. Lokale Korrelations- und Dichtefittingnäherungen wurden verwendet. Die fundamentalen Konzepte notwendig für das Verständnis von Coupled-Cluster-Theorie, Dichtefitting, lokaler Korrelation, allgemeinen Coupled-Cluster Eigenschaften und dem elektronischen g-Tensor werden diskutiert. Die berechneten g-Tensoren werden mit denen durch Coupled-Cluster Singles and Doubles, Dichtefunktionaltheorie und Experiment erhaltenen verglichen. Effizienz und Genauigkeit der Näherung wird untersucht. Ein detailierter Anhang beschreibt die diagrammatische Coupled-Cluster-Theorie sowie ihre Anwendung zur Herleitung der verwendeten Arbeitsgleichungen. Die in dieser Arbeit entwickelte Methode ermöglicht es, den elektronischen g-Tensor von ausgedehnten Systemen mit einer Methode, die nicht auf Dichtefunktionaltheorie basiert, quantitativ vorherzusagen. Damit ist sie ein wichtiger Schritt hin zur Entwicklung von niedrig skalierenden Coupled-Cluster-Methoden höherer Ordnung für diese Art von Problem. / This work presents an unrestricted coupled-cluster CC2 response method using local correlation and density fitting approximations for the calculation of first and second order properties with particular focus on the electronic g-tensor. The fundamental concepts related to coupled-cluster theory, density fitting, local correlation, general coupled-cluster properties and the electronic g-tensor are discussed. The calculated g-tensors are benchmarked against those obtained from coupled-cluster singles and doubles, density functional theory and experiment. Efficiency and accuracy of the approximations is investigated. A detailed appendix covers the fundamentals of diagrammatic coupled-cluster and its application to the derivation of the working equations. The method presented in this thesis enables the quantitative prediction of the electronic g-tensor of extended systems with a method other than density functional theory. It represents an important step towards the development of low-scaling higher order coupled-cluster methods for this type of problem.

Strongly correlated systems have been a major challenge for a long time in the field of theoretical chemistry. For such systems, the relevant portion of the Hilbert space scales exponentially, preventing efficient simulation on large systems. However, in many cases, the Hilbert space can be partitioned into clusters on the basis of strong and weak interactions. In this work, we mainly focus on an approach where we partition the system into smaller orbital clusters in which we can define many-particle cluster states and use traditional many-body methods to capture the rest of the inter-cluster correlations. This dissertation can be mainly divided into two parts. In the first part of this dissertation, the clustered ansatz, termed as tensor product states (TPS), is used to study large strongly correlated systems. In the second part, we study a particular type of strongly correlated system, correlated triplet pair states that arise in singlet fission. The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. We extend the incremental full configuration interaction originally proposed for a Slater determinant to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean-field reference TPS configuration and includes the correlation contribution of the excited TPSs using a many-body expansion. This method, named cluster many-body expansion (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from an RHF reference. The performance of the cMBE method is also tested on a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 1066 determinants for the exact FCI solution. Selected CI (SCI) using determinants becomes intractable for large systems with strong correlation. We introduce a method for SCI algorithms using tensor product states which exploits local molecular structure to significantly reduce the number of SCI variables. We demonstrate the potential of this method, called tensor product selected configuration interaction (TPSCI), using a few model Hamiltonians and molecular examples. These numerical results show that TPSCI can be used to significantly reduce the number of SCI variables in the variational space, and thus paving a path for extending these deterministic and variational SCI approaches to a wider range of physical systems. The extension of the TPSCI algorithm for excited states is also investigated. TPSCI with perturbative corrections provides accurate excitation energies for low-lying triplet states with respect to extrapolated results. In the case of traditional SCI methods, accurate excitation energies are obtained only after extrapolating calculations with large variational dimensions compared to TPSCI. We provide an intuitive connection between lower triplet energy mani- folds with Hückel molecular orbital theory, providing a many-body version of Hückel theory for excited triplet states. The n-body Tucker ansatz (which is a truncated TPS wavefunction) developed in our group provides a good approximation to the low-lying states of a clusterable spin system. In this approach, a Tucker decomposition is used to obtain local cluster states which can be truncated to prune the full Hilbert space of the system. As a truncated variational approach, it has been observed that the self-consistently optimized n-body Tucker method is not size- extensive, a property important for many-body methods. We explore the use of perturbation theory and linearized coupled-cluster methods to obtain a robust yet efficient approximation. Perturbative corrections to the n-body Tucker method have been implemented for the Heisenberg Hamiltonian and numerical data for various lattices and molecular systems has been presented to show the applicability of the method. In the second part of this dissertation, we focus on studying a particular type of strongly correlated states that occurs in singlet fission material. The correlated triplet pair state 1(TT) is a key intermediate in the singlet fission process, and understanding the mechanism by which it separates into two independent triplet states is critical for leveraging singlet fission for improving solar cell efficiency. This separation mechanism is dominated by two key interactions: (i) the exchange interaction (K) between the triplets which leads to the spin splitting of the biexciton state into 1(TT),3(TT) and 5(TT) states, and (ii) the triplet-triplet energy transfer integral (t) which enables the formation of the spatially separated (but still spin entangled) state 1(T...T). We develop a simple ab initio technique to compute both the triplet-triplet exchange (K) and triplet-triplet energy transfer coupling (t). Our key findings reveal new conditions for successful correlated triplet pair state dissociation. The biexciton exchange interaction needs to be ferromagnetic or negligible compared to the triplet energy transfer for favorable dissociation. We also explore the effect of chromophore packing to reveal geometries where these conditions are achieved for tetracene. We also provide a simple connectivity rule to predict whether the through-bond coupling will be stabilizing or destabilizing for the (TT) state in covalently linked singlet fission chromophores. By drawing an analogy between the chemical system and a simple spin-lattice, one is able to determine the ordering of the multi-exciton spin state via a generalized usage of Ovchinnikov's rule. In the case of meta connectivity, we predict 5(TT) to be formed and this is later confirmed by experimental techniques like time-resolved electron spin resonance (TR-ESR). / Doctor of Philosophy / The study of the correlated motion of electrons in molecules and materials allows scientists to gain useful insights into many physical processes like photosynthesis, enzyme catalysis, superconductivity, chemical reactions and so on. Theoretical quantum chemistry tries to study the electronic properties of chemical species. The exact solution of the electron correlation problem is exponentially complex and can only be computed for small systems. Therefore, approximations are introduced for practical calculations that provide good results for ground state properties like energy, dipole moment, etc. Sometimes, more accurate calculations are required to study the properties of a system, because the system may not adhere to the as- sumptions that are made in the methods used. One such case arises in the study of strongly correlated molecules. In this dissertation, we present methods which can handle strongly correlated cases. We partition the system into smaller parts, then solve the problem in the basis of these smaller parts. We refer to this block-based wavefunction as tensor product states and they provide accurate results while avoiding the exponential scaling of the full solution. We present accurate energies for a wide variety of challenging cases, including bond breaking, excited states and π conjugated molecules. Additionally, we also investigate molecular systems that can be used to increase the efficiency of solar cells. We predict improved solar efficiency for a chromophore dimer, a result which is later experimentally verified.

In dieser Arbeit wird eine unrestricted Coupled-Cluster CC2 Response-Methode für die Berechnung von Eigenschaften erster und zweiter Ordnung, mit dem elektronischen g-Tensor als Schwerpunkt, präsentiert. Lokale Korrelations- und Dichtefittingnäherungen wurden verwendet. Die fundamentalen Konzepte notwendig für das Verständnis von Coupled-Cluster-Theorie, Dichtefitting, lokaler Korrelation, allgemeinen Coupled-Cluster Eigenschaften und dem elektronischen g-Tensor werden diskutiert. Die berechneten g-Tensoren werden mit denen durch Coupled-Cluster Singles and Doubles, Dichtefunktionaltheorie und Experiment erhaltenen verglichen. Effizienz und Genauigkeit der Näherung wird untersucht. Ein detailierter Anhang beschreibt die diagrammatische Coupled-Cluster-Theorie sowie ihre Anwendung zur Herleitung der verwendeten Arbeitsgleichungen. Die in dieser Arbeit entwickelte Methode ermöglicht es, den elektronischen g-Tensor von ausgedehnten Systemen mit einer Methode, die nicht auf Dichtefunktionaltheorie basiert, quantitativ vorherzusagen. Damit ist sie ein wichtiger Schritt hin zur Entwicklung von niedrig skalierenden Coupled-Cluster-Methoden höherer Ordnung für diese Art von Problem. / This work presents an unrestricted coupled-cluster CC2 response method using local correlation and density fitting approximations for the calculation of first and second order properties with particular focus on the electronic g-tensor. The fundamental concepts related to coupled-cluster theory, density fitting, local correlation, general coupled-cluster properties and the electronic g-tensor are discussed. The calculated g-tensors are benchmarked against those obtained from coupled-cluster singles and doubles, density functional theory and experiment. Efficiency and accuracy of the approximations is investigated. A detailed appendix covers the fundamentals of diagrammatic coupled-cluster and its application to the derivation of the working equations. The method presented in this thesis enables the quantitative prediction of the electronic g-tensor of extended systems with a method other than density functional theory. It represents an important step towards the development of low-scaling higher order coupled-cluster methods for this type of problem.

People suffering from congenital anosmia show normal brain architecture although they do not have functional sense of smell. Some studies in this regard point to the changes in secondary olfactory cortex, orbitofrontal cortex (OFC), in terms of gray matter volume increase. However, diffusion tensor imaging has not been explored so far. We included 13 congenital anosmia subjects together with 15 controls and looked into various diffusion parameters like FA. Increased FA in bilateral OFC confirms the earlier studies reporting increased gray matter thickness. However, it is quite difficult to interpret FA in terms of gray matter volume. Increased FA has been seen with recovery after traumatic brain injury. Such changes in OFC point to the plastic nature of the brain.

Long-term cosmic-ray modulation studies using ab initio numerical modulation models require an understanding of the solar-cycle dependence of the heliospheric diffusion tensor. Such an understanding requires information as to possible solar-cycle dependences of various basic turbulence quantities. In this study, 1-minute resolution data for the N-component of the heliospheric magnetic field spanning from 1974 to 2012 is analysed using second-order structure functions constructed assuming a simple three-stage power-law frequency spectrum. This spectrum is motivated observationally and theoretically, and has an inertial, an energycontaining and a cutoff-range at small frequencies to ensure a finite energy density. Of the turbulence quantities calculated from 27-day averaged second-order structure functions, only the magnetic variance and the spectral level show a significant solar-cycle dependence, much less so the spectral index in the energy range. The spectral indices in the inertial range, as well as the turnover and cutoff scales do not appear to depend on the level of solar activity. The ratio of the variance to the square of the magnetic field also appears to be solar-cycle independent. These results suggest that the dominant change in the spectrum over several solar-cycles is its level. Comparisons of the results found in this study with relevant published observations of turbulence quantities are very favourable. Furthermore, when the magnetic variances and heliospheric magnetic magnitudes calculated in this study are used as inputs for theoretically motivated expressions for the mean free paths and turbulence-reduced drift lengthscale, clear solar-cycle dependencies in these quantities are seen. Values for the diffusion and drift lengthscales during the recent unusual solar minimum are found to be significantly higher than during previous solar minima. / MSc (Space Physics), North-West University, Potchefstroom Campus, 2015

Long-term cosmic-ray modulation studies using ab initio numerical modulation models require an understanding of the solar-cycle dependence of the heliospheric diffusion tensor. Such an understanding requires information as to possible solar-cycle dependences of various basic turbulence quantities. In this study, 1-minute resolution data for the N-component of the heliospheric magnetic field spanning from 1974 to 2012 is analysed using second-order structure functions constructed assuming a simple three-stage power-law frequency spectrum. This spectrum is motivated observationally and theoretically, and has an inertial, an energycontaining and a cutoff-range at small frequencies to ensure a finite energy density. Of the turbulence quantities calculated from 27-day averaged second-order structure functions, only the magnetic variance and the spectral level show a significant solar-cycle dependence, much less so the spectral index in the energy range. The spectral indices in the inertial range, as well as the turnover and cutoff scales do not appear to depend on the level of solar activity. The ratio of the variance to the square of the magnetic field also appears to be solar-cycle independent. These results suggest that the dominant change in the spectrum over several solar-cycles is its level. Comparisons of the results found in this study with relevant published observations of turbulence quantities are very favourable. Furthermore, when the magnetic variances and heliospheric magnetic magnitudes calculated in this study are used as inputs for theoretically motivated expressions for the mean free paths and turbulence-reduced drift lengthscale, clear solar-cycle dependencies in these quantities are seen. Values for the diffusion and drift lengthscales during the recent unusual solar minimum are found to be significantly higher than during previous solar minima. / MSc (Space Physics), North-West University, Potchefstroom Campus, 2015